Kosterlitz-Thouless transition and correlation function I’m studying Kosterlitz transition on this book:  https://tinymachines.weebly.com/uploads/5/1/8/8/51885267/kardar._statistical_physics_of_fields__2007_.pdf#page173 . At page 165 it says:” The gradient expansion applies to configuration that can be continuosly deformed to the uniformly ordered state”. My question is why? I know that at low temperature there are vortex antivortex pair (that can be transformed into ordered state) and an higher temperature vortex that cannot transformed into ordered state. 
 A: When doing gradient expansion, we are implicitly assuming $\theta(x)$ is a single valued function of $x$ and there is no discontinuity. Otherwise the gradient of $\theta$ diverges.
As the textbook says, any single-valued function can be continuously deformed to a uniformly ordered state. Here is how. Define a uniformly ordered state by $\theta_{uni}(x) = 0$ for all $x$. Then we can define a continuous deformation from $\theta_0(x)$ by
$$
\theta(x,\lambda) = (1-\lambda)\theta_0(x) - \lambda\theta_{uni}(x) = (1-\lambda)\theta_0(x)
$$ 
where $\lambda = 0$ correspond to the initial state and $\lambda=1$ corresponds to the final state. As long as $\theta_0(x)$ is smooth and continuous, this transformation is also smooth and continuous.
Vortex configurations cannot be described as a single valued function. For any single valued function $f(x)$ we have
$$
\int_C \nabla f(x) dx = 0
$$
for any closed loop C. However, for a loop C containing a vortex we have
$$
\int_C \nabla \theta(x) dx = 2\pi n
$$
where n is defined to be the vorticity of the vortex (+1 for vortex, -1 for anti vortex and so on). Thus, $\theta(x)$ cannot be a single valued function, and we need to put extra care in order to take vortices into account.
That explains why gradient approximation fails. The more interesting questions is why gradient approximation succeeds in some cases. As you have pointed out, at low temperature there are very few vortices present, so we can get away with ignoring their presence and using gradient expansion.
